Let \(\) , \(\)  and \(f\)  be as above.
Note that \(f(G)\)  is Zariski dense in the \(k\) -algebraic group \(f()\) . But generally, \(f(G)\)  is not onto the \(k\) -points \(f()(k)\) . For example let \(k=\) , take \(=\mathrm {SL}_{2}\)  and let \(f\)  be the central quotient map. Observe that
\(f(G)={2}{}\subsetneq {2}{}=f()(k)\) 

As seen in the remark above, one should be careful when dealing with \(k\) -points of quotient varieties, as these may not be \(k\) -points of a \(k\) -algebraic quotient.
We recall that in the case of a parabolic subgroup, or in the case of split solvable groups, the quotient of the \(k\) -points is equal to the \(k\) -points of the quotient group.
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A connected solvable \(k\) -group is called split if it admits a composition series by \(k\) -subgroups with successive quotients \(k\) -isomorphic to \(_m\)  or \(_a\) .

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If the field \(k\)  is perfect, then any connected unipotent group is \(k\) -split.

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Let \(\)  be a connected \(k\) -algebraic group and \(\)  a normal connected \(k\) -split solvable subgroup. Then the \(k\) -point algebraic map \(\pi \colon G \rightarrow /(k)\)  is surjective, i.e. \(G/A=/(k)\) .


Let \(\)  be a connected \(k\) -algebraic group. A subgroup \(P\le G\)  is called parabolic if
\(P=(k)\)  for some \(k\) -parabolic subgroup \(<\) .



Let \(\)  be a connected \(k\) -algebraic group, and \(P\le G\)  a parabolic subgroup. Then the center of the parabolic subgroup, \(Z(P)\) , equals the center of the group, \(Z(G)\) .
